Signed area in 2D
Starting with the unit square, elementary transformations shear, scale and reverse orientation. The final signed area gives the determinant.
Determinant from Elementary Matrices
The determinant of a square matrix is the signed scaling factor for area in 2D, volume in 3D and n-dimensional volume in general. These visuals show that scaling factor by building a transformed unit square or unit cube from elementary row-operation matrices.
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Signed volume in 3D
Starting with the unit cube, elementary transformations shear, scale and reverse orientation. The final signed volume gives the determinant.
Concept
Row replacement shears the square or cube and leaves signed area or volume unchanged. Row scaling multiplies signed area or volume by the scale factor. Row swapping reverses orientation, so the determinant changes sign.
Because an invertible matrix can be factored into elementary matrices, the determinant can be read as the accumulated signed scaling produced by those elementary transformations.
Structure
Matrix products act right to left, so the visual construction starts from the unit square or cube and applies inverse elementary matrices in sequence. The shape changes step by step, while the determinant records the product of the signed scaling factors.
Related visual: Determinant sign visualization
Related chapter: Determinant
Related work: GraphMath Linear Algebra
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